Preventive maintenance can reduce breakdowns and thecosts associated with them but is also costly when donefrequently. That is why substantial efforts have been

invested in minimizing the expected total cost due to failuresand preventive maintenance of industrial equipment (Dekker1996, Tan and Kramer 1997, Bäckert and Rippin 1985, Dekkerand Scarf 1998). In industry, most preventive-maintenanceapproaches include the use of fixed schedules, optimized inadvance for minimum cost. However, there are many situationsin which maintenance replanning is necessary to operate withas low cost as possible. For example, unexpected breakdownsforce the production unit to stop for emergency repair, and per-forming other maintenance tasks at the same time can save timeand money. In addition, the increasing use of condition moni-toring leads to more unpredictable maintenance events and aneed for dynamic planning.

In this article, we present the ideas behind the preventive-maintenance optimizer (PMOPT) tool used to optimize gas tur-bine maintenance schedules. We developed PMOPT for SiemensIndustrial Turbomachinery AB (SIT AB), one of the leading man-ufacturers of gas turbines of small and medium size. Gas tur-bines are used for power generation in various production facil-ities that often have high down time costs. A gas turbine in theform of a jet engine is depicted in figure 1. Axial flow gas tur-bines (including many industrial gas turbines) are constructedusing a compressor, which produces compressed air to a combus-tion chamber, where fuel is injected. The resulting mixture is

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SPRING 2010 21Copyright © 2010, Association for the Advancement of Artificial Intelligence. All rights reserved. ISSN 0738-4602

Searching for Gas Turbine Maintenance Schedules

Markus Bohlin, Kivanc Doganay, Per Kreuger, Rebecca Steinert, and Mathias Wärja

n Preventive-maintenance schedules occurringin industry are often suboptimal with regard tomaintenance coallocation, loss-of-productioncosts, and availability. We describe the imple-mentation and deployment of a software deci-sion support tool for the maintenance planningof gas turbines, with the goal of reducing thedirect maintenance costs and the often costlyproduction losses during maintenance downtime. The optimization problem is formallydefined, and we argue that the feasibility ver-sion is NP-complete. We outline a heuristicalgorithm that can quickly solve the problem forpractical purposes and validate the approach ona real-world scenario based on an oil produc-tion facility. We also compare the performanceof our algorithm with results from using integerprogramming and discuss the deployment of theapplication. The experimental results indicatethat down time reductions up to 65 percent canbe achieved, compared to traditional preventivemaintenance. In addition, the use of our tool isexpected to improve availability by up to 1 per-cent and to reduce the number of plannedmaintenance days by 12 percent. Compared toan integer programming approach, our algo-rithm is not optimal but is much faster and pro-duces results that are useful in practice. Our testresults and SIT AB’s estimates based on opera-tional use both indicate that significant savingscan be achieved by using our software tool,compared to maintenance plans with fixedintervals.

ignited, thereby increasing the volume and veloci-ty of the gas flow, in turn driving the turbine. Sincethe turbine is coupled to the compressor, the com-bustion cycle is sustained. A typical application ofindustrial gas turbines is offshore oil platforms,where power outage can cause an extremely highloss of revenue. In such applications, smallimprovements in terms of overall availability,which is one expected outcome of implementingcondition-based maintenance (CBM), have a sub-stantial positive effect on the total income for thecustomer.

Condition-based gas turbine maintenance,where component lifetime is dependent on factorssuch as load profile, quality of fuel, ambient tem-perature, and particle levels, is therefore becomingmore and more common. Although lifetime pre-dictions can sometimes be performed with highprecision, the predicted maintenance dates willsooner or later diverge from their estimatesdepending on on-site conditions.

The approach we present in this article providesthe means to quickly optimize maintenance whenunplanned events make the previous scheduleunsuitable. We use a rolled-out representation ofthe predicted future maintenance schedule andtake positive effects of co-allocation, overall avail-ability due to preventive maintenance, horizoneffects, and costs due to both maintenance andloss of production into account. Since proper riskanalysis and deterioration model identification isin many practical cases difficult to perform from

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scratch, maintenance intervals are often based onanalytical models and “best practice.” In PMOPT,we assume that a safe deadline for all maintenanceactivities exists, which simplifies the problem andmakes it easy to adapt already existing mainte-nance plans for use in PMOPT.

The contributions of this article include that we(1) precisely define the maintenance-schedulingproblem discussed, (2) argue that the planningproblem is NP-complete, (3) outline an algorithmthat can quickly solve the problem for practicalpurposes, (4) show results for a real-world scenario,(5) compare the results of our algorithm to theresults from using integer programming, and (6)discuss the implementation and deployment ofPMOPT.

Related Work Maintenance optimization has been an activeresearch area for a long time; Dekker (1996) gives agood overview of the different applications. Thework most closely related to our article is con-cerned with maintenance scheduling where eco-nomic dependencies between activities exist. Mostresearchers seem to use a common setup cost s forall maintenance activities performed at a singlemaintenance occasion. The cost saving for group-ing n maintenance activities together thenbecomes (n – 1)s. In this article, we propose a mod-el of indirect economic dependence between activ-ities, where the economic effect of joint execution

Intake Compression Combustion Exhaust

Air Inlet

Cold Section

Compression Combustion Chambers Turbine Exhaust

Hot Section

Figure 1. Schematics of a Gas Turbine.

Image taken from figure 14-1, Airplane Flying Handbook, U.S. Federal Aviation Administration 2004.

depends on the resulting down time of the main-tenance stop. The down time in turn depends pri-marily on the length of the activities at the stopand whether activities can be performed in paral-lel or not.

As an example of the former approach, Wilde-man, Dekker, and Smit (1997) discuss the type ofeconomic dependencies with one single shared set-up cost and in addition propose a polynomial solu-tion approach to the scheduling problem. Thepolynomial solution is optimal if the optimalgroups are always consecutive, that is, the activitiesoccur in the same order as their locally optimaldate. In the model proposed in this article, it mayvery well be optimal to group activities noncon-secutively if the earnings from doing so outweighthe costs. Van Dijkhuizen and van Harten (1997)propose a generalization to a tree-shaped setupstructure. Another approach is given by Yamayee,Sidenblad, and Yoshimura (1983), who use dynam-ic programming to optimize maintenance withrespect to equipment reliability, demand of gener-ating units, and maintenance cost. The main dif-ference compared to our work is that Yamayee etal. consider large-scale scheduling of power-gener-ating units, whereas we focus on single-unit multi-component maintenance, for which a detaileddown time model is beneficial.

Tan and Kramer (1997) consider opportunisticmaintenance in the chemical processing industry.Monte Carlo simulation is used to estimate costs,which allows a generic cost structure. In thisapproach, it is possible that an event with a lowprobability but a very high cost is not sampled andtherefore not taken into account. Another differ-ence compared to our work is that, in addition, weconsider nonhomogenous loss-of-productioncosts. The model we employ therefore allows us tooptimize maintenance for scenarios that includeperiods with both lower than normal down timecost and with a higher cost. This is important inmany industrial areas, for example, in the oil andgas industry. Marseguerra, Zio, and Podofillini(2002) also use Monte Carlo simulation and genet-ic algorithms and, in addition, consider otherproperties such as the number of maintenancetechnicians available. However, other economicdependencies between components are not con-sidered.

Background The common practice of gas turbine maintenanceplanning today is to base the schedules on equiva-lent operating hours (EOH) and cycles (number ofturbine restarts). In calculating EOH, we modifythe number of operating hours with factors forload, fuel quality, presence of water injection, and(to a limited extent) significant exhaust tempera-

ture differences. However, the model is notdetailed enough in how these variables are han-dled, and factors such as ambient air temperatureand pressure, rotational speeds, and more detailedoutlet temperatures are not included. Instead, theEOH calculations have built-in safety margins toaccommodate for variables not explicitly modeled.

To improve overall maintenance efficiency, newcalculations for estimating the remaining lifetimeof gas turbine components based on operation pro-file, environmental conditions, and condition data(obtained through inspections and sensors on thegas turbine) was developed by SIT AB. A lifetimeprediction tool, producing deterministic lifetimeestimates, was also developed. The lifetime esti-mates produced by the tool include relevant safetymargins. Therefore, changes in lifetime do notaffect risk levels negatively as long as the gas tur-bine is maintained within its predicted serviceperiod. In fact, risk levels are in many cases dra-matically reduced, since the lifetime predictiontool also detects and decreases maintenance inter-vals for gas turbines operating under conditionswith increased component wear (for example highload, high humidity, or low fuel quality).

Maintenance Activities In this article, we assume that the optimal mainte-nance intervals (t i

opt) for the individual compo-nents have already been calculated. We use theterm replacement of a component to representrepair to a state that is as good as new, whileinspections (including servicing, lubrication, on-condition, and failure-finding tasks) are considered“as-good-as-old” maintenance and do not affectcomponent lifetime.

Since replacements restore component lifetimefully, it is suboptimal to schedule inspections inde-pendently of the performed replacements. The sce-nario of independent inspections and replace-ments is shown in figure 2, where a total of fiveinspections are made. In contrast, the scenarioused in this article is shown in figure 3. Here, onlythree inspections are needed; the replacementsthat occur in between restore component status,and therefore, fewer inspections are needed. Notethat truly independent inspections can still bemodeled using a separate artificial component con-taining only inspection items.

The Gas Turbine Maintenance Process

In this section, we outline a workflow for the use ofgas turbine maintenance planning. The process isdependent on both online lifetime predictionsusing a prognostics tool and on-demand mainte-nance optimization using the PMOPT schedulingtool. The workflow shown in figure 4 starts with

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identification and specification of single and com-posed components and corresponding items. If anitem can belong to more than a single component,we either split the item into separate items for thecomponents in question or put the item in a gener-ic component schedule existing solely for that pur-pose.

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Independent of this activity, production plan-ning begins so that requirements from productioncan be used later in specifying constraints for themaintenance optimization. As an intermediatestep, initial lifetime predictions are carried outusing EOH and start-stop cycle estimates for com-ponents without damage accumulation algorithms

Time

Age

Overhaul Replacement

Inspections

Figure 2. Unrelated Replacements or Overhauls and Inspections of a Single Component.

Time

Age

Overhaul Replacement

Inspections

Inspection schedule

reset

Figure 3. Synchronized Replacements or Overhauls and Inspections, Resulting in the Elimination of Unnecessary Inspections (dotted).

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ComponentSpecification

Lifetime Predictionsusing prognostics

Lifetime Predictionsusing EOC/EOH

MaintenanceSpecification

Contract Specification OpportunitySpecification

Optimizationusing PMOpt

Corrective Actions /New Opportunities

Production Changes

RequirementsUpdate

Date Adjustment

Production Planning

New Contract

Significant LifetimeChanges

Start

TurbineRetired

Initial MaintenancePlanning

Production Revisions

Operational Mode

Operational Mode(Continuous Monitoring)

Small Changesin Lifetime

Figure 4. Workflow of Maintenance Planning using PMOpt.

� Minimum required availability. bj Base cost of opening opportunity j.vj Maximum work time at opportunity j.�j Date of an opportunity j. t[i] Maintenance date of item i.t i

opt Locally optimal time of maintenance for item i.t i

dl Deadline of item i, relative to pi.�pi Duration in phase q � 1..p for item i.t i

min Earliest possible date for item i.h Scheduling horizon. � Artificial item representing the schedule start. ci Cost of item i. � Artificial item representing the schedule end. kmax Maximum discrepancy parameter tried in LDS. t i

max Latest possible date for item i.li Loss of production cost at opportunity j.ℰ The set of head items. ℒ The set of tail items. obs(i) Predicate indicating whether item i is obsolete. pi Predecessor of item i.rj Resting time att opportunity j.t i

rt Release time of item i, relative to pi. si Terminator of item i.uj Total work time at opportunity j. A The number of working hours per day. b Maximum branching factor in the search tree. d Remaining depth of the search tree. k LDS discrepancy parameter. m Number of opportunities. n Number of items. p Number of phases. w Weight for cost due to unused lifetime.

or using prognostic tools for the other compo-nents.

From the lifetime predictions and productionrequirements, a maintenance specification is thenconstructed using PMOPT. It is important to pointout that as many requirements as possible are for-malized in the specification. At the same time, alist of known recurring and one-time maintenanceopportunities is made together with durations andestimates on the cost of down time during theopportunities. Further, a contract specification isneeded, formalizing the contract period and avail-ability requirements. Some of the contract require-ments, for example calendar constraints, may alsohave to be put in the maintenance specification; toavoid clutter, these requirements are not shown infigure 4.

Given the output of the process so far, PMOPT isnow ready to produce initial maintenance sched-ules. During production, PMOPT is used continual-ly and updated with the latest information as soonas it is available. For example, PMOPT needs toknow which part of the maintenance plan hasalready been executed and, consequently, which

items are still pending. Unexpected correctiveactions and maintenance opportunities are insert-ed manually together with suitable adjustments onthe remaining lifetime of the components affect-ed. Production changes mean that the future main-tenance opportunities, costs, and priorities change,and such changes must be fed into PMOPT to pro-duce relevant results. When the maintenance planof the gas turbine is changed for other reasons notpart of the CBM process, PMOPT still needs newmaintenance schedules, so that maintenance isplanned according to the current state of practice.

Scheduling Maintenance with Downtime Costs

We informally describe the maintenance schedul-ing with opportunities problem (MSOP) as theproblem of allocating n maintenance items to mdates (opportunities) for a set of independent com-ponents in a single unit and for a time period of hweeks, so that constraints on timeliness, mainte-nance duration, and total availability (due to pre-ventive maintenance) are satisfied. We assume thatmaintenance of any component means that theentire unit has to be shut down. The allocationshould minimize direct and indirect maintenancecosts, including spare parts, labor, and the value ofproduction that is lost due to maintenance. In therest of this section, we use terminology depicted inthe adjacent shaded box.

Each component has a cyclical schedule of arbi-trary length, consisting of replacement (Rx) andinspection (Ixy) items. The date of a replacementdepends on the previous replacement, whileinspections depend on the previous item regardlessof type. We assume that the given componentschedules are followed; deviations are taken intoaccount by updating the schedule data.

Duration and Downtime Models In maintaining a gas turbine, the work included ina single maintenance item can often be dividedinto different phases, for example the phases shut-down and cooling, dismantling, repair, reassembly,and startup, which are shown in figure 5.

To estimate work time at a maintenance stop,each maintenance item therefore has a durationspecification �i = ��1i, �2i, …, �pi� dividing work intop nonnegative work phases. All activities within asingle phase at a single stop are assumed to be ful-ly independent and can therefore be executed inparallel. In contrast, the phases themselves aredone in an orderly fashion. The total work time ujof a stop can thus be computed as the sum of themaximum work time in each block:

(1)uj i n

i jq p

qi=≤ ≤

≤ ≤∑ max

11

performed at

Δ

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An example is given in figure 5, where the fiveduration specifications �1 = �8, 0, 13, 0, 12�, �2 =�13, 7, 0, 7, 12�, �3 = �13, 7, 8, 17, 8�, �4 = �13, 15,7, 8, 12� and �5 = �13, 10, 12, 0, 12� are jointly per-formed at the same stop. The working time for thedifferent phases then becomes �13, 15, 13, 17, 12�,and the total work time at the stop is then 70hours.

Given the total work time at a stop, we can nowcompute the down time by adding resting time forthe maintenance crew. We assume that a workingday consists of A hours and that all calendar weeks(consisting of 6 working days) are alike. The restingtime at an opportunity consists of night and weekrest time and is computed as follows:

(2)

The total down time is the sum of working timeand resting time. Continuing our example in figure5 and assuming that A = 10, we obtain a restingtime of 108 hours and a total down time of 178hours.

Scheduling Model We assume that n maintenance items i have beenrolled out to cover weeks 1 to h (the horizon of the

rA

u

A

u

Au

j

j jj

=−( ) −

⎡

⎢⎢⎢

⎤

⎥⎥⎥+ −

⎡

⎢⎢⎢

⎤

⎥⎥⎥

>24 1 246

1 0

0

if

ootherwise

⎧

⎨

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

problem). The decision variable t[i] represents thedate of item i. The schedule end is modeled by theartificial item � at date h + 1, and the schedulestart is modeled by another artificial item � at date0. The possible allocation dates within the sched-ule are modeled by a set of opportunities j � 1..mwith dates �j and maximum work time vj.

The different timeliness constraints in the prob-lem are illustrated in figure 6 and can be expressedas follows. Each item i has a release time t i

rt and adeadline t i

dl relative to i’s predecessor pi. Each itemalso has an optional earliest and latest date of exe-cution (t i

min and t imax). After a replacement, a se -

quence of inspections should follow before a newreplacement is made. Similarly, the scheduleshould contain enough replacements to cover thescheduling period, but extra replacements shouldnot be taken into account. We call rolled-out itemsthat do not have to be executed obsolete items. Forthis purpose, each item i has a terminator si thatmakes i obsolete if i is done later or at the samedate as si. For simplicity, we force obsolete items tobe allocated to the same date as their terminator.Formally, we define the predicate obs(i), with themeaning that item i is made obsolete by its termi-nator si, as follows.

obs(i) � t[i] = t[si] (3)

Replacements always have � as their terminator,which implies that they are obsolete when they are

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Figure 5.

Duration and down time model, where work is divided into phases in which activities can be executed in parallel. Resting time is thenadded. We assume that Mondays are spent travelling.

not performed before the problem horizon h. Fig-ure 6 illustrates relative timeliness constraintsbetween pairs of tasks as well as predecessor andterminator relationships in a fictional schedule.The set of items that are first in the schedule foreach component is called the set of head items, andis denoted ℰ. All head items have � as their pred-ecessor.

To ensure that the gaps after sequences of itemsare not too large, we use special items representingthe end of such sequences. We call such items tailitems. The set of tail items ℒ consists of (1) the lastreplacement for each component and (2) the lastitem in each inspection sequence. To ensure thatthe generated schedules cover the scheduling peri-od, we force all tail items to be obsolete. The dead-line constraints then ensure that the gaps at theend of an inspection or replacement sequence aresmaller than required.

Each item also has an item cost ci consisting ofwork and material cost. In addition, the value ofproduction per hour at an opportunity j is denot-ed lj. We also use a base cost bj for opening upopportunity j. The base cost is associated with

shared setup costs not related to the down time ofthe opportunity and includes some of the costs forshutting down and restarting the gas turbine, trav-el expenses, and other shared costs that cannot bemodeled using material, work, or down time costs.Minimum availability is specified using the param-eter � (where 0 ≤ � ≤ 1). The total availability isdefined as the time not spent on preventive main-tenance divided by the total available time.

The constraints can now be formally stated.

Each item i should be allocated to a date less thanor equal to its deadline.

�i � 1..n : = t[i] ≤ t[pi] + t idl (4)

Each item has to respect its absolute allocationinterval.

�i � 1..n : = t imin ≤ t[i] ≤ t i

max (5)

Each tail item has to be obsolete.

�i � ℒ : obs(i) (6)

Each nontail item i should be either obsolete orallocated to a date larger than its offset.

�i � 1..i \ℒ : obs(i) � t[i] ≤ t[pi] + t irt (7)

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Figure 6.

Dependencies (top) and relative timeliness constraints (bottom) between different maintenance items of a component.

For each opportunity j, the work time uj of j must belower than the maximum work time vj.

�j � 1..m : uj ≤ vj (8)

The availability of the plan should be greater thanthe minimum availability �.

(9)

The objective of the optimization problem is tominimize the cost function f, defined as follows:

(10)

The first term is the maintenance cost of all itemsperformed before the schedule horizon, the secondterm is the indirect cost due to loss of production,and the third term is the total of the shared costsdue to initiation of maintenance (base costs).

Complexity Let FMSOP be the problem of determining whethera feasible maintenance schedule exists. In this sec-tion, we argue that FMSOP is NP-complete byshowing that (1) FMSOP is in NP, since it has apolynomial-time verification algorithm (Cormenet al. 2001), and (2) that there is a polynomial-timereduction from the bin packing problem (BPP)(Coffman, Garey, and Johnson 1997) to FMSOP.The objective of BPP is to pack items i � {1, …, n}of given sizes ai into as few bins (with fixed capac-ity V) as possible. The used capacity of a bin is com-puted as the sum of the weights of the items in thebin. The decision variant of BPP answers the ques-tion whether a packing for any given number ofbins m exists.

Given a candidate solution to FMSOP (that is anassignment of dates to the maintenance items), wecan verify the constraints on structure and timeli-ness by simply testing equations 4–7 for the givendates of the item and its predecessor and termina-tor. This verification can be done in O(n) time. Them capacity constraints in equation 8 can easily beverified by accumulating the items allocated to theopportunities in time O(np + m), where m is thenumber of opportunities and p is the number ofphases. The availability constraint in equation 9can also, in a similar fashion, be verified in O(np +m) time. The procedure outlined above is polyno-mial, and therefore FMSOP is in NP.

We can translate a given BPP into a FMSOP by hav-ing m opportunities, each opportunity j having date�j = j and capacity vj = V. Let the horizon h = m + 1.Each BPP item i is translated into a FMSOP replace-ment item i with � as predecessor, 0 as release time,m as deadline, t i

min = 0 and t imax = h. The duration �qi

= ai if q = i and 0 otherwise, that is, the duration(weight) of an item is always put in a unique phasein �i. All items i have an artificial item n + i as ter-minator, which in turn have release time 1, dead-

f c l u r bij m

i

j j jj m

jj m

i

t( )= + +( )+∈

¬ ( )

∈ ∈

∃ ∈

∑ ∑1 1 1… … …

obs 11…n i j:t[ ]=

∑δ

17 24

11⋅

+ ≤ −( )∈∑ u r hj

j mj

…

α

line h +1, t imin = 0, t i

max = h +1, � as terminator andarbitrary duration. By definition, the tail items,being replacements, have to occur at �, which isoutside the schedule. Let the minimum availability� = 0. The transformed problem corresponds direct-ly to BPP, since (1) each BPP item is represented byan FMSOP replacement, (2) each BPP bin is repre-sented by an FMSOP opportunity with unique dateand equal capacity, and (3) the total down time ofan opportunity is computed as the sum of the itemdurations at that opportunity, since all durationsare in unique working phases, which correspondsdirectly to the sum of the weights of items in a binin BPP. All other constructs of FMSOP are disabledand therefore do not constrain the solution, andtherefore, BPP is a special case of FMSOP.

If we could find a solution to the transformedFMSOP using a polynomial-time algorithm, wecould then use that algorithm to solve BPP (whichis NP-complete [Garey and Johnson 1979]) in poly-nomial time. This property together with the pre-vious conclusion that FMSOP is in NP implies thatFMSOP is NP-complete.

Efficient polynomial-time approximations existfor the bin-packing problem (Coffman, Garey, andJohnson 1997). However, MSOP differs in objectivefrom BPP and has complicating side constraintsthat are missing in BPP. For example, in MSOP,each opportunity (date) can have a different capac-ity, base cost, and down time cost. In BPP, a bin isdefined only by its capacity, which is also uniform.Another difference is that items in MSOP can par-tially overlap within an opportunity due to thework time model used. These properties make bin-packing approaches inapplicable to MSOP. It is cur-rently an open issue whether polynomial-timeapproximation schemes exist for MSOP.

A Tool for Maintenance Scheduling The optimization software consists of two separateprograms that communicate using files; PMOPT -GUI and MAINTOPT. The schedule and related infor-mation are considered to be a project and are storedin a project file. A typical user would load a previ-ously created project file directly after startingPMOPT. PMOPT-GUI makes it possible to edit theproject file, and it immediately displays the effectsof edits, such as costs and availability. Edits includechanging lifetime estimates, adding or deletingcomponents and items, and moving or copyingitems within and between components. Whenev-er the user requests an optimization of the currentmaintenance plan, PMOPT-GUI produces a rolled-out representation of the specification, which ispassed on to the optimizer. As soon as MAINTOPT

finishes, the solution file is read back into PMOPT-GUI and shown to the user.

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Optimization Algorithm The requirement for the optimization algorithm isthat it should be able to produce maintenanceschedules within a limited time to be used interac-tively. Many industrial problems can be solvedusing tree search methods, especially if guidingheuristics are available. For example, best-firstsearch methods such as A* search have been suc-cessful on many problems (Nilsson 1971; Hart,Nilsson, and Raphael 1972). However, A* search isreliant on the availability of a good admissibleheuristic, and if no such heuristic is available, A*search will on some problems use up too muchmemory and time to be practically useful. Depth-first-based search methods avoid the memoryissues of breadth-first search and A* but can easilyget stuck in unproductive areas of the search treewhen the heuristic fails. Limited discrepancysearch (LDS) addresses this problem (Harvey andGinsberg 1995; Korf 1996). The basic idea of LDS isto use depth-first search guided by a heuristic butto allow a specified number of “discrepant” choic-es disagreeing with the heuristic. The number ofdiscrepant choices allowed in each path from rootto leaf is the discrepancy parameter k.

The basic LDS procedure introduced by Harveyand Ginsberg (1995) only works on binary searchtrees, although the authors also discuss extensionsto nonbinary problems. Algorithm 1 shows such anextension, also modified to continue searching fora best possible solution as measured by an objectivefunction f, which is infinitely valued for the emptynode NIL. In the original paper, it is discussed

whether all discrepant choices emanating from aspecific node should be treated equally, that is,counted as “using up” a discrepancy of one, orwhether each further step away from the heuristicshould be counted as using up one more discrepantchoice. In algorithm 1, the latter view is taken. Fig-ure 7 shows the unique paths explored for eachchoice of k in a tree with branching factor 3.

Korf (1996) modified LDS so that it only gener-ates paths with exactly k discrepancies. This modi-fication is done by keeping track of the remainingdepth d, pruning branches for which d ≤ k. Whilemodifying LDS to nonbinary trees, a similarimprovement can be done if the maximum branch-ing factor b is known. At most d discrepant choicescan be made in a subtree of depth d, each choiceusing up at most b − 1 discrepancies. We can there-fore prune choices i where (b – 1)(d – 1) + i < k, or inother words, start with choice number max(0, k – (b– 1)(d – 1)). We assume that the function call SUC-CESSORS(node) returns a list of feasible successornodes in increasing order of heuristic value.

In the LDS procedure we use, maintenance itemsare assigned in order of increasing deadline, andthe value-selection heuristic picks opportunities inincreasing cost order, with a bias for late opportu-nities. In essence, the heuristic used is defined as

(11)

where t[i � �j] is the result of assigning item i toopportunity j given assignment t, w is a balancingweight, and t i

opt is the original (optimal) mainte-nance interval for component i. t i

opt is differentfrom t i

dl in that t idl may be changed due to con-

sumed lifetime of component i. Note that variabledomains are pruned using interval propagation(Lhomme 1993). In our experiments, we foundthat a maximum discrepancy of kmax = 2 gave over-all good performance. We used w = 1 as the weightvalue to favor late assignments, and the defaultoptimization time is set to 30 seconds, which ismore than enough for the problem instances wehave tried.

Development and Deployment Manual planning is the norm in the gas turbinefield, and before PMOPT and the lifetime predictiontool, SIT AB did not have any manual or automat-ic procedures for improving maintenance sched-ules. Instead, it used a standard schedule equal to20,000 operating hours subject to a constant levelof degradation at a standard pace for all compo-nents in the gas turbine. The initial effort at SIT ABfocused on developing more accurate lifetime pre-dictions, and the need for optimization of mainte-nance schedules first became apparent when thetechnology was driven to the point that the total

h i f i wcp t

tj j ii i j

i

t tt

δ δδ⎡

⎣⎤⎦( )= ⎡

⎣⎤⎦( )+ [ ]+ −dl

opt

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LDS-PROBE(node, k, d, b)(1) if LEAF(node) then return node(2) �suc0, …, sucb–1� ← SUCCESSORS(node)(3) bt ← NIL

(4) for i = max(0, k – (b – 1)(d – 1)) to min(b – 1, k)(5) bt ← argminf (bt, LDS-PROBE(suci, k – i, d, b))(6) return bt

LDS(node, d, b, kmax)(1) res ← NIL

(2) for k ← 0 to kmax

(3) res ← argminf (res, LDS-PROBE(node, k, d, b))(4) return res

Algorithm 1. Limited Discrepancy Search for Nonbinary Optimization Problems with Objective f.

effect of component lifetime extensions could beevaluated.

We were first approached by SIT AB regardingmaintenance-scheduling optimization during thesummer of 2006 at an international conferencerelated to condition monitoring. This first contactresulted in a sequence of meetings during theautumn at which we discussed and evaluated thefeasibility of the project idea. At this time point,we had already developed the core maintenance-optimization engine (MAINTOPT). However, MAIN-TOPT was in its infancy, and we soon became awarethat we had to extend it with side constraints andobjective function terms that we had not previ-ously considered. One example is the availability

constraints and the focus on down time as a criti-cal parameter, which was not implemented inMAINTOPT at that time. However, being able todemonstrate our ideas helped a lot during the ini-tial meetings.

Before starting the PMOPT development process,a third-party commercial product for maintenanceoptimization was evaluated at SIT AB. One of themain problems with the third-party product wasthat it could not model important properties of thegas turbine planning problem, such as seasonalvariations and usage profiles for different parame-ters like load, particle levels, and environmentalfactors. More importantly, the evaluated tool, andmost other generic maintenance-optimization

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(a) k = 0 (b) k = 1

(c) k = 2 (d) k = 3 (e) k = 4

(f) k = 5 (g) k = 6

Figure 7. Unique Explored Edges for Different Discrepancy Values in a Tree with Branching Factor 3.

tools, use costs based on statistics. In reality, it iscommon that the statistically optimal point oflowest cost is not practically feasible due to lack ofdata and the need for large safety margins. Theconsequences of some types of failures, includingthe possibility of loss of life, are under these cir-cumstances too severe to be modeled and used as abasis for planning. In addition, for a complexmachine such as a gas turbine, it is impractical toidentify all possible failures, the corresponding sta-tistical distributions, and all consequences andassociated costs for each failure. Instead of havingtoo many estimates, we decided that a safe dead-line for each maintenance item was a more rea-sonable alternative.

Throughout the project, a total of five peoplefrom the Swedish Institute of Computer Sciencewere directly involved. We had two main contactpersons at SIT AB and decided to directly involveseveral site managers for evaluation. SIT AB wasalso heavily involved in the specification anddevelopment throughout the project, and thisform of close collaboration allowed us to designaccurate models of the application and was in ouropinion a main factor behind the successful proj-ect outcome.

First Versions In November 2006, we received a spreadsheet con-taining an early draft specification of the problemto be solved. The spreadsheet showed some ideasregarding calculations on down time and mainte-nance-activity packaging, and we decided to devel-op a prototype from the draft specification. Theprototype was nothing more than a simple graph-ical front end to MAINTOPT without any interac-tion. Nonetheless, it served the purpose of show-ing the feasibility of the project proposal well.After this prestudy and basic demonstration, webegan discussions regarding the project economyand deliverables in early 2007. Soon after that thecontracts were signed and development started.We finished the first release (version 0.9) in mid-April 2007. Due to time pressure, the first versionwas more of a prototype than a mature softwareproduct. With many test releases in between, ver-sion 1.0 was finally shipped in June 2007. Theglobal CBM project was however not fully opera-tional at that time, and version 1.0 was thereforemainly used for evaluation and as a platform forfurther development.

From experience with the first releases, we soonrealized that changes in the optimization enginewere rather straightforward to implement. Howev-er, software maintenance and extensions that pri-marily affected the management of the problemmodel was much more time-consuming. One ofthe biggest problems was to implement functional-ity to manage the maintenance schedule while ver-

ifying model consistency under the entire set ofuser actions possible. For example, changes in themaintenance schedule made after running the gasturbine for some time needed to be synchronizedwith the already laid-out maintenance schedule upto the current time point. Other areas that we choseto put more work into than estimated were themodels of work time, application security, licens-ing, management of gas turbine maintenance proj-ects, and user accounts and rights management.

Second Version We made several changes to the basic design ofPMOPT in the second phase of the project to sim-plify the software maintenance of the applicationand facilitate future extensions. Rewriting the coreof the application from scratch was perhaps thelargest change, but we also made some significantchanges in the search algorithm, which took someeffort. In the beginning, we implemented MAIN-TOPT as a pure branch-and-bound algorithm basedon A* search (Russell and Norvig 2003). However,after extensive experimentation with samplemaintenance projects we realized that the A*search spent too much time and memory explor-ing high-level decisions in the search tree andfailed to find feasible solutions quickly. Sinceresponsiveness was one of the main criteria ofPMOPT, we resorted to experimentation withheuristics and after a while added the LDS proce-dure as a first stage of the algorithm to quickly findfeasible solutions. Lately, we have completelyremoved the A* search from MAINTOPT after observ-ing that the heuristics were not nearly powerfulenough. Under these conditions, the A* search didnot explore the search tree to its maximum depthwithin a reasonable time, and therefore, no feasi-ble solutions were found. The ability to produce areasonably good solution fast was more importantthan optimality, and we therefore chose LDS as theprimary search algorithm.

With major changes to the GUI and improvedheuristics, we released version 2.0 in March 2008,two months in advance of its deadline due to themuch improved core design, which helped speed upthe implementation of new features. Since then, wehave made more improvements and shipped a newrelease in August the same year. The latest release(version 2.4) was shipped in November 2008.

Deployment at SIT AB During the development of PMOPT, we becameincreasingly aware that a planning tool of this typeis not easily deployed. First of all, key personnelhad to be trained in condition-based maintenanceand in how to use PMOPT. During the developmentof the first version, we continuously discussed sug-gestions and ideas for the usability enhancementof the software. Before deployment could begin,

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we also had to develop suitable business modelsand finish evaluating current technology. In addi-tion, data-acquisition routines, working processes,and suitable information flows needed to be estab-lished. Therefore, PMOPT was not deployed inoperational use until early 2008 after the release ofversion 2.0.

The greatest benefit of using PMOPT seems tocome from the ability to reoptimize maintenanceafter unexpected stops have occurred. The softwarehas been used operatively for five months byemployees at SIT AB to help plan maintenance fortwo gas turbine operators (end-user customers).PMOPT is fully operational in planning mainte-nance for the first operator and is used for valida-tion and testing purposes of the global CBM strat-egy and the technology involved for the second.We test the planning software to gain feedbackfrom practical experience using both PMOPT andthe lifetime prediction algorithms. SIT AB plans tohave four or five people working with PMOPT for10–15 different operational contracts within a fewyears. Today, the tool is used interactively byexperts in extended lifetime maintenance. Whenmore people are starting to use PMOPT on a largerscale, changes and upgrades will likely be necessaryas the user requirements on the applicationemerge.

Application Maintenance and Support We performed software maintenance of PMOPT ondemand when bug reports were filed, which hap-pened mostly from our main contact people at SITAB after new releases had been shipped. Naturally,we got most bug reports just after the delivery ofversion 1.0. Overall, larger improvements weremostly related to the GUI and the usability of thesystem. Current users have direct contact with usand are able to ask questions as well as requestchanges. During the development, we improvedour understanding of the domain substantially,and we gradually implemented several new fea-tures in the problem models. SIT AB explicitlyrequested some changes, while we judged otherchanges to be necessary to make the code base easyto maintain. As an example, we changed the spec-ification of the optimization model several times.Specifically, after a request from SIT we updatedthe work time model to more accurately capturethe real duration, down time, and cost of a main-tenance opportunity. In addition, the team alsoproposed other changes regarding the model ofdependencies between maintenance items and thehandling of obsolete items.

Estimated and Measured Benefits In this section, we evaluate PMOPT using conditiondata and operational plans for a gas turbine used

by a customer in the oil and gas industry. The costof one engineer man-hour is roughly 120 USD, andthe loss-of-production cost is estimated to 12,000USD per hour. The results of the evaluation arebased on the predicted cost savings produced byPMOPT. The turbine under consideration has 17components with individual schedules. We use astandard maintenance schedule for the site as com-parison. We modeled the critical components inthe gas generator stage for which lifetime data wasavailable (compressor turbine guide vanes andblades) and analyzed the components in a prog-nostics tool to determine suitable inspection inter-vals. The average increase in inspection period was116 percent, and replacements for the critical com-ponents were not necessary, since their predictedlifetime became longer than the standard mainte-nance-contract length of 15 years. We describedthe scenario in more detail in a previous paper(Bohlin et al. 2009).

SIT AB has done a partial validation of theobtained lifetimes in that a reference gas turbine,having operated under the same conditions, wasdismantled after a standard maintenance intervalof 20,000 operating hours and thoroughly inspect-ed. The analysis showed that the accumulateddamage was significantly less than predicted usingthe standard EOH calculations. However, final val-idation has to wait until one or more reference gasturbines have been dismantled after a longer oper-ational period.

We performed the evaluation on two scenarios.The first scenario had a three-week seasonal stopduring the summer, when maintenance could bedone without any negative effects on production.Such opportunities for maintenance are commonin practice. In the second scenario, no such favor-able opportunities existed. In both scenarios, a lowbase cost was associated with all maintenancestops, and high costs were associated with loss ofproduction. The schedules resulting from runningPMOPT were analyzed with regard to cost of pro-duction losses and maintenance costs. PMOPT wasset to run for at most 30 seconds. In all examplestried, at most 10 hours of work per day wereallowed before a resting period.

Results Table 1 shows simulated results on availability, rel-ative maintenance load, and down time for theeight different cases and a brand new gas turbine.The rows “EOH” and “Progn” correspond to plan-ning maintenance at the last possible date, as spec-ified using standard EOH calculations and theprognostics tool, respectively. This approach min-imizes direct maintenance costs while ignoringother costs. The rows marked “EOH opt” and“Progn opt” correspond to optimizing mainte-nance using PMOPT.

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Results are reported for availability due to pre-ventive maintenance (“Avail”), maintenance costs(“Maint index”), and productive down time (“DTdays”). Zero-cost down time corresponds to noproduction losses and is, therefore, in contrast tothe theoretical model used in the optimization,not considered as down time in the results. Main-tenance costs are expressed using an index. In it,100 represents the cost of doing maintenance withthe intervals from the standard schedule. In table1, the results of using the standard schedule aretypeset in boldface. Note that these results are sub-

optimal when low-cost opportunities are present. As can be seen in table 1, better lifetime esti-

mates had a significant impact on maintenancecosts, availability, and down time. Adding the opti-mization of maintenance using PMOPT yields evenbetter results and increases direct maintenancecosts slightly. These results are to be expected,since production losses in this case are very costlyand optimization is done with regard to both lossof production costs and direct maintenance costs.When zero-cost maintenance opportunities wereavailable, productive down time was reduced to

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With Seasonal Stop Without Seasonal Stop

Availability Percentage

Maintenance index

Down-time days

Availability Percentage

Maintenance index

Down-time days

EOH 97.60 100 131 97.60 100 131

EOH opt 99.99 109 0.42 98.15 120 101

Progn 98.20 61 98 98.20 61 98

Progn opt 100.0 62 0 98.81 75 65

With Seasonal Stop Without Seasonal Stop

Availability Percentage

Maintenance index

Down-time days

Availability Percentage

Maintenance index

Down-time days

EOH 97.60 100 131 97.60 100 131

EOH opt 99.99 109 0.42 98.15 120 101

Progn 98.20 61 98 98.20 61 98

Progn opt 100.0 62 0 98.81 75 65

With Seasonal Stop Without Seasonal Stop

Availability Percentage

Maintenance index

Down-time days

Availability Percentage

Maintenance index

Down-time days

EOH 95.26 121 259 95.26 121 259

EOH opt 99.56 133 24.0 97.49 149 137

Progn 96.03 79 217 96.03 79 217

Progn opt 99.79 82 11.6 98.35 85 90

Table 1. Predicted Results of Maintenance Optimization for a New Gas Turbine.

Table 2. Predicted Results of One Maintenance Optimization for a Gas Turbine with Randomly Chosen History for Its 17 Components.

With Seasonal Stop Without Seasonal Stop

Difference Percentage

Gap Percentage

Time Difference Percentage

Gap Percentage

Time

New Turbine

EOH opt –6.1 0 40 minutes – ∞ 8 hours

Progn opt –1.1 0 27 minutes +93 75.6 8 hours

Used Turbine

Progn –23.6 1.79 8 hours – ∞ 8 hours

Progn opt – 0.6 0.95 8 hours – ∞ 8 hours

Table 3. Comparison of Results between CPLEX 9.0 and PMOPT.

Negative numbers mean that CPLEX produced a lower cost than PMOPT.

zero or almost zero by planning maintenance forthe available opportunities. Table 1 also shows thatfor the schedule with no advantageous opportuni-ties, down time was reduced by more than 50 per-cent using PMOPT and prognostics.

Table 2 shows predicted results for the eight dif-ferent cases and gas turbine already in use, simu-lated by setting the already-used lifetimes of thegas turbine components to a random numberdrawn from a uniform distribution between zeroand the maintenance interval for the component.As expected, table 2 shows higher costs and loweravailability than table 1 due to a more spread outmaintenance need. Using a prognostics tool andPMOPT in this scenario also yielded significantresults. In the experiments, down time wasreduced by 65 percent for a schedule with noadvantageous opportunities, compared to the cur-rent state of practice. In the case where seasonalopportunities were present, down time wasreduced by 95 percent.

Comparison with Integer Programming To investigate how far away from the optimum theresults from PMOPT were, we formulated MSOP asan integer linear-programming problem. We usedILOG CPLEX 9.0 on a mainframe computer with a2.2 GHz dual core AMD Opteron CPU and 8 GB ofRAM to solve the problem. The total run time foreach case was limited to 8 hours. Although run-ning an algorithm for such a long time is not suit-able for our needs, the comparison still gives usvaluable insight in where PMOPT can be improved.In contrast, PMOPT was run on a laptop with a 1.6GHz Intel CPU for 30 seconds in each case.

Results for the eight different cases in the previ-ous sections are compared in table 3. Diff gives therelative difference between the best found cost forPMOPT and CPLEX, with negative values indicatingthat CPLEX found a better solution than PMOPT.The Gap column gives the relative optimality gap(distance to the relaxed optimum) as returned byCPLEX, with higher values indicating that the gapis larger. The gap is infinite if no feasible solutionwas found within 8 hours. The Time columnreports CPU runtime for proving optimality, with acutoff at 8 hours.

For the two cases with a new turbine and sea-sonal stops, CPLEX was able to find the exact opti-mal solution (indicated by a gap value of 0). Forthe two cases with a used turbine and seasonalstops, CPLEX had found better solutions thanPMOPT when 8 hours had passed, with a quitesmall optimality gap. While CPLEX producesslightly better results for cases with lifetimes fromthe prognostics tool (Progn), the instances withstandard EOH lifetimes appear to benefit more sig-nificantly. It is notable that CPLEX reports a resultthat is more than 23 percent better than PMOPT in

the case with EOH lifetimes and seasonal stops.However, when there are no seasonal stops, CPLEXcannot find a solution even close to the result fromPMOPT within 8 hours.

Conclusions and Future Work

We described the development and deployment ofan opportunity-based maintenance-planning tool,PMOPT, specifically designed to fit the purpose ofimproving the maintenance schedules for gas tur-bines from SIT AB. The goal was to reduce bothdirect maintenance costs and production losses.Thanks to close collaboration with key personnelat SIT AB, we gained important insights into indus-trial maintenance planning, which allowed us todesign and implement the maintenance-planningtool. We believe that the working process we usedhas contributed greatly to the successful deploy-ment of PMOPTS.

We formally described and characterized thescheduling problem as NP-complete, and discusseda heuristic algorithm for solving it. Our experi-ments using real-world data for lifetime predic-tions showed a potential for significantly reduceddown time (with up to 65 percent) and costs.Experiments with integer programming gave evengreater gains but at the cost of much longer solu-tion times. Expected effects in practical use includelarge availability improvements and preventive-maintenance reductions of up to 12 percent.Future plans include fleet-level planning andlabor-resource optimization and scheduling, inclu-sion of stochastic deterioration models and expect-ed costs due to corrective maintenance, and appli-cation to other domains. We are also consideringinvestigating solution sensitivity with regard toparameter changes.

References Bäckert, W., and Rippin, D. W. T. 1985. The Determina-tion of Maintenance Strategies for Plants Subject toBreakdown. Computers and Chemical Engineering 9(2):113–126.

Bohlin, M.; Wärja, M.; Holst, A.; Slottner, P.; andDoganay, K. 2009. Optimization of Condition-BasedMaintenance for Industrial Gas Turbines: Requirementsand Results. Paper no. GT2009-59935 presented at theASME Gas Turbine Technical Congress and Exposition,June 8–12, Orlando, FL.

Coffman, E. G.; Garey, M. R.; and Johnson, D. S. 1997.Approximation Algorithms for Bin Packing: A Survey.Chapter 2 in Approximation Algorithms for NP-Hard Prob-lems, ed. D. Hochbaum, 46–93 Boston: PWS PublishingCompany.

Cormen, T. H.; Leiserson, C. E.; Rivest, R. L.; and Stein,C. 2001. Introduction to Algorithms, second edition.Cambridge, MA: The MIT Press.

Dekker, R. 1996. Applications of Maintenance Optimiza-

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Nilsson, N. J. 1971. Problem-Solving Methods in ArtificialIntelligence. New York: McGraw-Hill.

Russell, S., and Norvig, P. 2003. Artificial Intelligence: AModern Approach, second edition. Engelwood Cliffs, NJ:Prentice Hall.

Tan, J. S., and Kramer, M. A. 1997. A General Frameworkfor Preventive Maintenance Optimization in ChemicalProcess Operations. Computers and Chemical Engineering21(12): 1451–1469.

Van Dijkhuizen, G., and van Harten, A. 1997. OptimalClustering of Frequency-Constrained Maintenance Jobswith Shared Set-Ups. European Journal of OperationsResearch 99(3): 552–564.

Wildeman, R. E.; Dekker, R.; and Smit, A. C. J. M. 1997. ADynamic Policy for Grouping Maintenance Activities.European Journal of Operations Research 99(3): 530–551.

Yamayee, Z.; Sidenblad, K.; and Yoshimura, M. 1983. AComputationally Efficient Optimal Maintenance Sched-uling Method. IEEE Transactions on Power Apparatus andSystems AS-102(2): 330–338.

Markus Bohlin is a researcher at the Swedish Institute ofComputer Science. He has more than 10 years of experi-ence with industrially-motivated research. In 2009, hesuccessfully defended his Ph.D. thesis on combinatorialoptimization in industrial computer systems. Hisresearch is focused on real-world combinatorial opti-mization with applications in maintenance, softwareengineering and logistics..

Kivanc Doganay is a researcher at the Swedish Instituteof Computer Science. He holds a M.Sc. in computer sci-ence from Mälardalen University, and was a full-timedeveloper exclusively for the late releases of PMOpt. Cur-rently, he works with maintenance optimization in therailway sector and industrial data analysis.

Per Kreuger is a senior researcher at the Swedish Insti-tute of Computer Science (SICS) specializing in appliedcombinatorial optimization. He received his Ph.D. at theComputing Science Department of Gothenburg Univer-sity in 1995 and has since then worked mainly with prac-tical combinatorial problems using constraint program-ming and mathematical programming techniques withintransportation, networking, and related areas. Recentachievements related to AI includes cochairing and edit-ing the proceedings of the Scandinavian AI conference2008.

Rebecca Steinert was employed in 2006 as a researcherby the Swedish Institute of Computer Science (IndustrialApplications and Methods lab) in Kista, Sweden. In 2008,she received a M.Sc. in computer science from KTH inStockholm, Sweden and is now a Ph.D. student at theSchool of Computer Science and Communications, KTH.Her research is focused on statistical methods for fault-handling in industrial applications and networked sys-tems.

Mathias Wärja has been a specialist in CBM for rotatingequipment at Siemens Industrial Turbomachinery ABsince 2004. In 2005, he received his Ph.D. in industrialinformation and control systems from KTH, Stockholm.

tion Models: A Review and Analysis. Reliability Engineer-ing and System Safety 51(3): 229–240.

Dekker, R., and Scarf, P. 1998. On the Impact of Optimi-sation Models in Maintenance Decision Making: TheState of the Art. Reliability Engineering and System Safety60(9): 111–119.

Garey, M. R., and Johnson, D. S. 1979. Computers andIntractability: A Guide to the Theory of NP-Completeness.San Francisco: W. H. Freeman.

Hart, P. E.; Nilsson, N. J.; and Raphael, B. 1972. Correc-tion to ”A Formal Basis for the Heuristic Determinationof Minimum Cost Paths.” SIGART Bulletin (37): 28–29.

Harvey, W. D., and Ginsberg, M. L. 1995. Limited Dis-crepancy Search. In Proceedings of the Fourteenth Interna-tional Joint Conference on Artificial Intelligence, 607–615.San Francisco: Morgan Kaufmann Publishers.

Korf, R. E. 1996. Improved Limited Discrepancy Search.In Proceedings of the Eighth Innovative Applications of Arti-ficial Intelligence Conference, 286–291. Menlo Park, CA:AAAI Press.

Lhomme, O. 1993. Consistency Techniques for NumericCSPs. In Proceedings of the Thirteenth International JointConference on Artificial Intelligence, 232–238. San Francis-co: Morgan Kaufmann Publishers.

Marseguerra, M.; Zio, E.; and Podofillini, L. 2002. Condi-tion-Based Maintenance Optimization by Means ofGenetic Algorithms and Monte Carlo Simulation. Relia-bility Engineering and System Safety 77(2): 151–165.

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